Optimization Formula

The Formula

Find critical points by solving f'(x) = 0. Second derivative test: if f''(c) < 0, local max; if f''(c) > 0, local min.

When to use: Find where the function hits its peaks (maxima) and valleys (minima) by finding where the slope is zero.

Quick Example

Maximize area of rectangle with fixed perimeter: take derivative, set to zero, solve.

Notation

Critical point: c where f'(c) = 0 or f'(c) is undefined. Local max/min at c.

What This Formula Means

The process of using derivatives to systematically find maximum or minimum values of a function over a domain.

Find where the function hits its peaks (maxima) and valleys (minima) by finding where the slope is zero.

Formal View

f has a local maximum at c if \exists \delta > 0 : f(x) \leq f(c)\; \forall x \in (c - \delta, c + \delta). Necessary condition (Fermat's theorem): if f is differentiable at an interior extremum c, then f'(c) = 0. Second derivative test: f'(c) = 0 \land f''(c) < 0 \implies local max; f'(c) = 0 \land f''(c) > 0 \implies local min.

Worked Examples

Example 1

easy
Find the local maximum and minimum values of f(x) = x^3 - 3x + 2.

Solution

  1. 1
    Find f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x-1)(x+1).
  2. 2
    Set f'(x) = 0: critical points at x = 1 and x = -1.
  3. 3
    Find f''(x) = 6x. At x = -1: f''(-1) = -6 < 0, so local maximum.
  4. 4
    At x = 1: f''(1) = 6 > 0, so local minimum.
  5. 5
    Evaluate: f(-1) = -1 + 3 + 2 = 4 (local max); f(1) = 1 - 3 + 2 = 0 (local min).

Answer

Local maximum: f(-1) = 4; local minimum: f(1) = 0
The second derivative test classifies critical points: negative second derivative means concave down (local max), positive means concave up (local min). Always evaluate the function at the critical point to get the actual max/min value.

Example 2

hard
A farmer has 200 m of fencing to enclose a rectangular field. What dimensions maximize the enclosed area?

Common Mistakes

  • Forgetting to check endpoints of a closed interval: the absolute max or min often occurs at a boundary, not at a critical point where f'(x) = 0.
  • Assuming every critical point is a maximum or minimum: f'(c) = 0 could also be an inflection point (e.g., f(x) = x^3 at x = 0) — use the first or second derivative test to classify.
  • Setting up the wrong function to optimize in word problems: misidentifying what quantity to maximize or minimize, or writing the constraint equation incorrectly.

Why This Formula Matters

Practical applications: minimize cost, maximize profit, optimize design.

Frequently Asked Questions

What is the Optimization formula?

The process of using derivatives to systematically find maximum or minimum values of a function over a domain.

How do you use the Optimization formula?

Find where the function hits its peaks (maxima) and valleys (minima) by finding where the slope is zero.

What do the symbols mean in the Optimization formula?

Critical point: c where f'(c) = 0 or f'(c) is undefined. Local max/min at c.

Why is the Optimization formula important in Math?

Practical applications: minimize cost, maximize profit, optimize design.

What do students get wrong about Optimization?

Check endpoints too—max/min might be at boundaries, not where derivative = 0.

What should I learn before the Optimization formula?

Before studying the Optimization formula, you should understand: derivative.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Derivatives Explained: Rules, Interpretation, and Applications →
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Related Concepts

Derivative

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Derivative Formula